In this work we consider the ultimate dynamics of the Kirschner-Panetta model which was created for studying the immune response to tumors under special types of immunotherapy. New ultimate upper bounds for compact invariant sets of this model are given, as well as su cient conditions for the existence of a positively invariant polytope. We establish three types of conditions for the nonexistence of compact invariant sets in the domain of the tumor-cell population. Our main results are two types of conditions for global tumor elimination depending on the ratio between the proliferation rate of the immune cells and their mortality rate. These conditions are described in terms of simple algebraic inequalities imposed on model parameters and treatment parameters. Our theoretical studies of ultimate dynamics are complemented by numerical simulation results.
Alessio Cardillo (IPHES)
The synchronization of coupled oscillating systems is a phenomenon that has received considerable attention from the scientific community given its wide range of applications. The pattern of interactions among the oscillators -- usually encoded as a complex network — plays a crucial role in promoting the onset of the synchronized state and, over the years, several studies have investigated the emergence of synchronization in populations of oscillators arranged as a network.
Despite the amount of studies hitherto made, all the approaches rely on the hypothesis that the variation of the state for an oscillator, which is a fundamental requirement to attain synchronization, is costless. Yet, it seems reasonable to assume that the alteration of an oscillator's state involves an adjustment cost that, in turns, reverberates on the dynamics. The introduction of such cost leads to the formulation of a dichotomous scenario. In such situation, an oscillator may decide to pay the cost necessary to alter its state and make it more similar to that of the others or, alternatively, keep it unaltered waiting that the others will adjust their states to its own. The former behavior can be viewed as an act of cooperation while the latter as one of defection; both constitute the basic action profiles of a Prisoner's Dilemma game.
Complex networks play a key role also in the emergence of cooperation and, in particular, the presence of hubs in scale-free networks bolsters such phenomenon. Thus, it is worth investigating the underlying mechanisms responsible for the onset of synchronization in systems where the oscillators correspond to the nodes of a network and can decide to cooperate, by synchronizing their states with those of their neighbors, or not. This leads to a coevolutionary approach where the synchronization dynamics and the evolution of cooperation are intertwined together. Coevolutionary approaches are the natural extension of the actual models to attain a better description of complex systems. In this talk, I will present a coevolutionary model of Kuramoto oscillators playing an evolutionary game which strategically decide whether to cooperate or defect upon the evaluation of their payoff received during the synchronization stage. Finally, the emergence of both cooperation and synchronization is studied in three different topologies, namely: Erdös-Rényi random graphs, Random Geometric Graphs and Barabási-Albert scale-free networks.
Jon Chapman (University of Oxford)
M.C. Escher is known as the mathematician's (and hippie's) favourite artist. But why? And was Escher, a man who claimed he knew no mathematics, really a mathematical genius? In this lecture Jon Chapman not only show why Escher has won the artistic and mathematical hearts of mathematicians, but also why his art is inspiring both artists and mathematicians today, as captured in Jon's brilliant updating of Escher's "Picture Gallery" to the new mathematics building in Oxford.
Bill Thompson (Max Plank Institute for Psycholinguistics)